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1 Purpose

F08XPF (ZGGESX) computes the generalized eigenvalues, the generalized Schur form S,T and, optionally, the left and/or right generalized Schur vectors for a pair of n by n complex nonsymmetric matrices A,B.

Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.

3 Description

The generalized Schur factorization for a pair of complex matrices A,B is given by

A=QSZH, B=QTZH,

where Q and Z are unitary, T and S are upper triangular. The generalized eigenvalues, λ, of A,B are computed from the diagonals of T and S and satisfy

Az=λBz,

where z is the corresponding generalized eigenvector. λ is actually returned as the pair α,β such that

λ=α/β

since β, or even both α and β can be zero. The columns of Q and Z are the left and right generalized Schur vectors of A,B.

Optionally, F08XPF (ZGGESX) can order the generalized eigenvalues on the diagonals of S,T so that selected eigenvalues are at the top left. The leading columns of Q and Z then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.

F08XPF (ZGGESX) computes T to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.

The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in RCONDE1 and RCONDE2 respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in RCONDV1 and RCONDV2. See Section 4.11 of Anderson et al. (1999) for further information.

On exit: ALPHAj/BETAj, for j=1,2,…,N, will be the generalized eigenvalues. ALPHAj and BETAj,j=1,2,…,N are the diagonals of the complex Schur form S,T. BETAj will be non-negative real.

Note: the quotients ALPHAj/BETAj may easily overflow or underflow, and BETAj may even be zero. Thus, you should avoid naively computing the ratio α/β. However, ALPHA will always be less than and usually comparable with A in magnitude, and BETA will always be less than and usually comparable with B.

On exit: if INFO=0, the real part of WORK1 contains a bound on the value of LWORK required for optimal performance.

21: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08XPF (ZGGESX) is called.

If LWORK=-1, a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if N=0, LWORK≥1;

if SENSE='E', 'V' or 'B', LWORK≥max1,2×N,2×SDIM×N-SDIM;

otherwise LWORK≥max1,2×N.

Note:2×SDIM×N-SDIM≤N×N/2. Note also that an error is only returned if LWORK<max1,2×N, but if SENSE='E', 'V' or 'B' this may not be large enough. Consider increasing LWORK by nb, where nb is the optimal block size.

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08XPF (ZGGESX) is called.

If LIWORK=-1, a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if SENSE='N' or N=0, LIWORK≥1;

otherwise LIWORK≥N+2.

25: BWORK(*) – LOGICAL arrayWorkspace

Note: the dimension of the array BWORK
must be at least
1 if SORT='N', and at least max1,N otherwise.

After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy SELCTG=.TRUE.. This could also be caused by underflow due to scaling.

INFO=N+3

The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

such that the eigenvalues of A,B for which λ<6 correspond to the top left diagonal elements of the generalized Schur form, S,T. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.

Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.